Interfacial Line Energy of a Topological Phase

TL;DR

The paper investigates nucleation in a two-dimensional system where a topological fermionic field (a Chern insulator) is coupled to a scalar Ising order parameter. Using a BHZ-like lattice model, it shows that topological edge modes on a nucleating droplet introduce a quantum correction to the interfacial line energy, quantified by Δσ, which raises the critical nucleus size from n_c^{cl} to n_c^{Q} and yields a tunable enhancement γ = (1 + Δσ/σ_cl)^2. The authors combine analytic estimates of edge-state contributions with exact and effective Monte Carlo simulations to demonstrate that topological edge states dominate the interfacial energy in the topological phase, while bulk contributions in the trivial phase are non-universal. This work links boundary topological physics to classical nucleation, with potential implications for domain formation in correlated topological materials and for understanding quantum corrections to phase transitions.

Abstract

In interacting topological systems, Landau-like order parameters interplay with the band topology of fermions. The physics of domain formation in such systems can get significantly altered due to the presence of topological fermions. In this work we show that coupling a topological fermionic field to a scalar field can drastically modify the nucleation processes. We find that existence of non-trivial fermionic boundary modes on the nucleating droplets of the scalar field leads to substantial quantum corrections to the interface energy. This leads to an increase in the size of the critical nucleus beyond which unrestricted droplet growth happens. To illustrate the phenomena we devise a minimal model of fermions in a Chern insulating system coupled to a classical Ising field in two spatial dimensions. Using a combination of analytic and numerical methods we conclusively demonstrate that topological phases have a characteristic quantum correction to the interfacial line energy. Apart from material systems, our work opens up a host of questions regarding the impact of fermionic topological terms on classical phase transitions and associated criticality.

Figures (14)

• Figure 1: Topological droplet: (a) Free-energy density (blue curves) for a two-dimensional classical Ising model coupled to topological fermions in magnetic field $h$ and temperature $T$ where $C$ is Chern number and arrow shows sudden quench. (b) $C=1$ droplet within a metastable region of $C=-1$. (c) A droplet of size $n$ either grows or shrinks depending on a critical size. This critical size for classical Ising system ($n_{c}^{\rm{cl}}$) increases to $n_{c}^{\rm {Q}}$ when coupled to topological fermions due to droplet edge-modes leading to additional interfacial line energy.
• Figure 2: Interface energy and critical cluster-size: (a) Change of ground-state energy $\Delta E$ for fermionic system with $\sqrt{n}$ where $n$ is cluster-size. (Inset) Correction to interfacial line energy $\Delta \sigma$ obtained from slope of linear fit (see Eq. \ref{['eq_dE']}) in (a) as a function of $\kappa$, where dotted lines denote quantum critical points $\kappa=-2,0,2$. (b) $\gamma=n_{c}^{\rm{Q}}/n_{c}^{\rm{cl}}$ (see Eq. \ref{['eq_gamma']}) with $\kappa$ and $J$ at $T=0$, where $n_{c}^{\rm{cl}}$ is critical cluster-size for classical situation ($\kappa=0$). Here, number of unit cells is $L^{2}=4096$. In topological situation ($0<|\kappa|<2$), $n_{c}^{\rm{Q}}>n_{c}^{\rm{cl}}$ and relative enhancement $\gamma$ increases with decreasing $J$.
• Figure 3: Local magnetization in coupled spin-fermion Monte Carlo simulation: Local magnetization $s_{x,y}$ for unit cell with position $(x,y)$ at times $t=0,20,40$ (in units of Monte Carlo steps per site) in a typical realization of the coupled spin-fermion exact Monte Carlo simulation (see Eq. \ref{['eq_diff_qme']} and Eq. \ref{['eq_qm_mc']}) for $\kappa=1.2$. Here, the initial cluster-size with up-spins is $n=441$. The parameters chosen are: $J=0.2$, $T=1/\beta=0.33$ (where $T_{c} \approx 0.45$), $h=0.02$ and number of unit cells in square lattice $L^{2}=4096$. We observe the growth of the cluster.
• Figure 4: Local magnetization in a typical realization: Local magnetization $s_{x,y}$ for unit cell with position $(x,y)$ at times $t=0, 20, 100$ (in units of Monte Carlo steps per site) in a typical realization for $\kappa=0$ and $\kappa=1.2$ where initial cluster-sizes with up-spins are $n=441$, $n=225$ and $n=81$. The parameters chosen are: $J=0.2$, $T=1/\beta=0.33$ (where $T_{c} \approx 0.45$), $h=0.02$. Number of unit cells in square lattice is $L^{2}=4096$. The cluster with initial size $n=441$ grows in both the decoupled ($\kappa=0$) and topological ($\kappa=1.2$) situations, while the cluster with $n=81$ shrinks in both the situations. When $n=225$, we observe the growth of the cluster for $\kappa=0$ and the shrinkage of the cluster for $\kappa=1.2$.
• Figure 5: Critical cluster-size and droplet growth: (a) $\nu$ as a function of cluster-size $n$ for classical ($\kappa=0$), topological ($\kappa=1.2$) and trivial ($\kappa=2.5$) situations (see Eq. \ref{['eq_nu']}). $\nu =0$ denotes the critical sizes which are $n_{c}^{\rm{cl}}=145$, $n_{c}^{\rm{Q}}=304$, $n_{c}^{\rm{Q}}=158$ respectively. While in topological situation $n_{c}^{\rm{Q}}>n_{c}^{\rm{cl}}$, in trivial situation $n_{c}^{\rm{Q}} \approx n_{c}^{\rm{cl}}$. (b) Initial and final average local magnetization $\langle s_{x,y} \rangle$ when clusters of sizes, $n=441$ ($n>n_{c}^{\rm{Q}}$), $n=225$ ($n_{c}^{\rm{Q}}>n>n_{c}^{\rm{cl}}$), $n=81$ ($n_{c}^{\rm{cl}}>n$) are evolved. The final time $t=100$ is in units of MC steps per site. Upper (lower) panel shows $\kappa=0$ ($\kappa=1.2$) and $J=0.2$, $T=1/\beta=0.33 < T_{c} \approx 0.45$, $h=0.02$, $L^{2}=4096$. Averaging is performed over $10^3$ simulations. For classical ($\kappa=0$) situation, the cluster grows when $n>n_{c}^{\rm{cl}}$ but in topological situation ($\kappa=1.2$), growth occurs only when $n>n_{c}^{\rm{Q}}$.